Instructions
Welcome to the Numerical Archaeology Lab! Your mission is to decode and analyze large natural numbers using two ancient but powerful techniques: Expanded Notation and Prime Factorization. Follow the steps below precisely.
- Complete all required questions in Sections 1 and 2.
- Use exponents (powers) in all final answers, as shown in the examples.
- Attempt the Challenge Question in Section 3 for bonus credit.
Section 1: Place Value Decoded (Expanded Notation using Powers of 10)
Expanded notation breaks a number down based on the value of each digit, represented by powers of 10 (e.g., $10^0 = 1$, $10^1 = 10$, $10^2 = 100$, etc.).
Task: Write the following natural numbers in expanded notation using powers of 10. You must include terms for zero placeholders.
| Number | Expanded Notation (using Powers of 10) |
|---|---|
| Example: 5,409 | (5 10³) + (4 10²) + (0 10¹) + (9 10⁰) |
| 832 | |
| 1,750 | |
| 20,416 | |
| 300,071 | |
| 1,900,200 |
Section 2: Prime Power Signatures
Every natural number greater than 1 has a unique prime factorization. This means expressing the number as a product of prime numbers, written using exponents.
Reminder: The first few prime numbers are 2, 3, 5, 7, 11, 13, 17...
Task: Find the Prime Factorization for the following numbers and write the final result using exponent notation.
| Number | Prime Factorization (Exponent Form) |
|---|---|
| Example: 72 | *2³ 3²* (Because 72 = 8 9) |
| 60 | |
| 150 | |
| 243 | |
| 500 | |
| 1,125 |
Section 3: The Ultimate Decode (Synthesis Challenge)
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Reverse Engineering: A number is represented by the following expanded notation: $$(4 \cdot 10^5) + (2 \cdot 10^3) + (6 \cdot 10^1) + (3 \cdot 10^0)$$ What is the standard form of this number? (Hint: Watch out for the missing powers of 10).
Standard Form:
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Challenge Question (Optional): Consider the number $N = 360$. a) Write $N$ in Expanded Notation using Powers of 10:
b) Write $N$ as a Prime Power Signature (Prime Factorization):
Answer Key
Section 1: Place Value Decoded (Expanded Notation using Powers of 10)
| Number | Expanded Notation (using Powers of 10) |
|---|---|
| 832 | (8 10²) + (3 10¹) + (2 * 10⁰) |
| 1,750 | (1 10³) + (7 10²) + (5 10¹) + (0 10⁰) |
| 20,416 | (2 10⁴) + (0 10³) + (4 10²) + (1 10¹) + (6 * 10⁰) |
| 300,071 | (3 10⁵) + (0 10⁴) + (0 10³) + (0 10²) + (7 10¹) + (1 10⁰) |
| 1,900,200 | (1 10⁶) + (9 10⁵) + (0 10⁴) + (0 10³) + (2 10²) + (0 10¹) + (0 * 10⁰) |
Section 2: Prime Power Signatures
| Number | Prime Factorization (Exponent Form) |
|---|---|
| 60 | 2² 3¹ 5¹ |
| 150 | 2¹ 3¹ 5² |
| 243 | 3⁵ |
| 500 | 2² * 5³ |
| 1,125 | 3² * 5³ |
Section 3: The Ultimate Decode (Synthesis Challenge)
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Reverse Engineering: Standard Form: 402,063
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Challenge Question: $N = 360$ a) Expanded Notation: *(3 10²) + (6 10¹) + (0 10⁰) b) Prime Power Signature: 2³ 3² 5¹* (Since 360 = 8 9 * 5)